Embedded newform 210.3.w.a.47.2

• Label: 210.3.w.a.47.2
• Level: $210$
• Weight: $3$
• Character: 210.47
• Analytic conductor: $5.722$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	210.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.72208555157\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			47.2
Character	\(\chi\)	\(=\)	210.47
Dual form			210.3.w.a.143.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-2.84148 + 0.962278i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-4.69598 - 1.71691i) q^{5} +(-2.35455 - 3.52932i) q^{6} +(-1.00349 + 6.92770i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(7.14804 - 5.46859i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 32 q^{2} - 6 q^{3} - 12 q^{5} + 4 q^{7} - 128 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(71\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	+	1.36603i	0.183013	+	0.683013i
							\(3\)	−2.84148	+	0.962278i	−0.947161	+	0.320759i
\(5\)	−4.69598	−	1.71691i	−0.939196	−	0.343381i
							\(7\)	−1.00349	+	6.92770i	−0.143356	+	0.989671i
\(11\)	8.28471	−	4.78318i	0.753155	−	0.434834i								−0.0736777	−	0.997282i	\(-0.523474\pi\)
							0.826833	+	0.562448i	\(0.190140\pi\)
\(13\)	−9.08652	−	9.08652i	−0.698963	−	0.698963i	0.265224	−	0.964187i	\(-0.414554\pi\)
							−0.964187	+	0.265224i	\(0.914554\pi\)
\(17\)	6.55954	−	24.4806i	0.385856	−	1.44003i	−0.450958	−	0.892545i	\(-0.648918\pi\)
							0.836814	−	0.547488i	\(-0.184416\pi\)
\(19\)	10.4928	−	18.1740i	0.552250	−	0.956526i	−0.445861	−	0.895102i	\(-0.647103\pi\)
							0.998112	−	0.0614237i	\(-0.0195641\pi\)
\(23\)	−33.1729	+	8.88865i	−1.44230	+	0.386463i	−0.893338	−	0.449384i	\(-0.851643\pi\)
							−0.548962	+	0.835848i	\(0.684977\pi\)
\(29\)	−6.08119			−0.209696			−0.104848	−	0.994488i	\(-0.533436\pi\)
							−0.104848	+	0.994488i	\(0.533436\pi\)
\(31\)	7.22620	−	4.17205i	0.233103	−	0.134582i	−0.378900	−	0.925438i	\(-0.623697\pi\)
							0.612003	+	0.790856i	\(0.290364\pi\)
\(37\)	−14.2075	+	3.80688i	−0.383986	+	0.102889i	−0.445648	−	0.895208i	\(-0.647027\pi\)
							0.0616620	+	0.998097i	\(0.480360\pi\)
\(41\)	−55.3266			−1.34943			−0.674714	−	0.738079i	\(-0.735733\pi\)
							−0.674714	+	0.738079i	\(0.735733\pi\)
\(43\)	57.2947	−	57.2947i	1.33243	−	1.33243i	0.429248	−	0.903187i	\(-0.358779\pi\)
							0.903187	−	0.429248i	\(-0.141221\pi\)
\(47\)	−74.4838	+	19.9579i	−1.58476	+	0.424636i	−0.940396	−	0.340081i	\(-0.889545\pi\)
							−0.644366	+	0.764717i	\(0.722879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	210.3.w.a.47.2	yes	64
3.2	odd	2		210.3.w.b.47.12	yes	64
5.3	odd	4		210.3.w.b.173.7	yes	64
7.3	odd	6	inner	210.3.w.a.17.5	✓	64
15.8	even	4	inner	210.3.w.a.173.5	yes	64
21.17	even	6		210.3.w.b.17.7	yes	64
35.3	even	12		210.3.w.b.143.12	yes	64
105.38	odd	12	inner	210.3.w.a.143.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.w.a.17.5	✓	64	7.3	odd	6	inner
210.3.w.a.47.2	yes	64	1.1	even	1	trivial
210.3.w.a.143.2	yes	64	105.38	odd	12	inner
210.3.w.a.173.5	yes	64	15.8	even	4	inner
210.3.w.b.17.7	yes	64	21.17	even	6
210.3.w.b.47.12	yes	64	3.2	odd	2
210.3.w.b.143.12	yes	64	35.3	even	12
210.3.w.b.173.7	yes	64	5.3	odd	4